On the additive bases problem in finite fields

We prove that if $G$ is an Abelian group and $A_1,\ldots,A_k \subseteq G$ satisfy $m A_i=G$ (the $m$-fold sumset), then $A_1+\ldots+A_k=G$ provided that $k \ge c_m \log n$. This generalizes a result of Alon, Linial, and Meshulam [Additive bases of vector spaces over prime fields. J. Combin. Theory Ser. A, 57(2):203--210, 1991] regarding the so called additive bases.


Introduction
Let p be a fixed prime, and let Z n p denote the n-dimensional vector space over the field Z p . Given a multiset B with elements from Z n p , let S(B) = b∈S b S ⊆ B . The set B is called an additive basis if S(B) = Z n p . Jaeger, Linial, Payan, and Tarzi [JLPT92] made the following conjecture and showed that if true, it would provide a beautiful generalization of many important results regarding nowhere zero flows. In particular the case p = 3 would imply the weak 3-flow conjecture, which has been proven only recently by Thomassen [Tho12]. Conjecture 1. [JLPT92] For every prime p, there exists a constant k p such that the union (with repetitions) of any k p bases for Z n p forms an additive basis.
Let us denote by k p (n) the smallest k ∈ N such that the union of any k bases for Z n p forms an additive basis. In [ALM91] two different proofs are given to show that k p (n) ≤ c p log n, where here and throughout the paper the logarithms are in base 2. The first proof is based on exponential sums and yields the bound k p (n) ≤ 1 + (p 2 /2) log 2pn, and the second proof is based on an algebraic method and yields k p (n) ≤ (p − 1) log n + p − 2. As it is observed in [ALM91], it is easy to construct examples showing that k p (n) ≥ p, and in fact, to the best of our knowledge, it is quite possible that k p (n) = p.
Let G be an Abelian group, and for A, B ⊆ G, define the sumset Theorem 2 (Main theorem). Let G be a finite Abelian group and suppose that A 1 , . . . , A 2K ⊆ G satisfy We present the proof of Theorem 2 in Section 2. While it is quite possible that Conjecture 1 is true, the following example shows that its generalization, Theorem 2, cannot be improved beyond Θ(log log |G|) even when m = 2.
Example 3. Let n = 2 k and for i = 1, . . . , k, let C i ⊆ Z 2 i p be the set of vectors in Z 2 i p \ { 0} in which the first half or the second half (but not both) of the coordinates are all 0's. Note that It follows from On the other hand a simple induction shows that for j ≤ k, Remark 4. Theorem 2 in particular implies that k p (n) ≤ 2(p − 1) ln n + 2(p − 1) ln log p, and k 3 (n) ≤ 2 log n + 2. Note that for p > 3, the algebraic proof of [ALM91] provides a slightly better constant, however unlike the theorem of [ALM91], Theorem 2 can be applied to the case where p is not necessarily a prime.

Proof of Theorem 2
The proof is based on the Plünnecke-Ruzsa inequality.

Quasi-random Groups
While Example 3 shows that the bound of Θ(log log |G|) is essential in Theorem 2, for certain non-Abelian groups, it is possible to achieve the constant bound similar to what is conjectured in Conjecture 1. A finite group G is called D-quasirandom if all non-trivial unitary representations of G have dimension at least D. The terminology "quasirandom group" was introduced explicitly by Gowers in the fundamental paper [Gow08] where he showed that the dense Cayley graphs in quasirandom groups are quasirandom graphs in the sense of Chung, Graham, and Wilson [CGW89]. The group SL 2 (Z p ) is an example of a highly quasirandom group. The so called Frobenius lemma says that SL 2 (Z p ) is (p − 1)/2-quasirandom. This has to be compared to the cardinality of this group, |SL 2 (Z p )| = p 3 − p. The basic fact that we will use about the quasirandom groups is the following theorem of Gowers (See also [Tao15, Exercise 3.1.1]).
We will also need the noncommutative version of Ruzsa's inequality.
Lemma 7 (Ruzsa inequality). Let A, B, C ⊆ G be finite subsets of a group G. Then Proof. The claims follows immediately from fact that by the identity ac −1 = ab −1 bc −1 , every element ac −1 in AC −1 has at least |B| distinct representations of the from xy with (x, y) ∈ (AB −1 ) × (BC −1 ).
Finally we can state the analogue of Theorem 2 for quasi-random groups.